MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-nfc Structured version   Visualization version   GIF version

Definition df-nfc 2740
Description: Define the not-free predicate for classes. This is read "𝑥 is not free in 𝐴". Not-free means that the value of 𝑥 cannot affect the value of 𝐴, e.g., any occurrence of 𝑥 in 𝐴 is effectively bound by a "for all" or something that expands to one (such as "there exists"). It is defined in terms of the not-free predicate df-nf 1701 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
Assertion
Ref Expression
df-nfc (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Detailed syntax breakdown of Definition df-nfc
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 cA . . 3 class 𝐴
31, 2wnfc 2738 . 2 wff 𝑥𝐴
4 vy . . . . . 6 setvar 𝑦
54cv 1474 . . . . 5 class 𝑦
65, 2wcel 1977 . . . 4 wff 𝑦𝐴
76, 1wnf 1699 . . 3 wff 𝑥 𝑦𝐴
87, 4wal 1473 . 2 wff 𝑦𝑥 𝑦𝐴
93, 8wb 195 1 wff (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
Colors of variables: wff setvar class
This definition is referenced by:  nfci  2741  nfcr  2743  nfcd  2746  nfceqdf  2747  nfnfc1  2754  nfnfc  2760  nfnfcALT  2761  drnfc1  2768  drnfc2  2769  dfnfc2  4390  dfnfc2OLD  4391  nfnid  4823  bj-nfnfc  32047  bj-nfcf  32112
  Copyright terms: Public domain W3C validator